Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 1481–1510

# A Complete Spectral Analysis of Generalized Gribov–Intissar’s Operator in Bargmann Space

• Jean-Karim Intissar
Article

## Abstract

In the Bargmann representation, we study mathematically rigorously some interesting spectral properties of generalized Gribov–Intissar’s operator: $$\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda } = \lambda ' a^{*^{p+1}} a^{p+1}} {+ \mu a^{*^{p}}a^{p} + i\lambda a^{*^{p}}(a + a^{*})a^{p}\, (p = 0, 1, 2\ldots )}$$ where $$a^{*}$$ and a are the creation and annihilation operators; $$[a, a^{*} ] = \mathbb {I }$$. $$(\lambda ', \mu , \lambda ) \in \mathbb {R}^{3}$$ are respectively the four coupling, the intercept and the triple coupling of Pomeron and $$i^{2} = - 1$$. Firstly, the domain of the operator is defined precisely and it is shown that the minimal and the maximal version of the operator are identical; here as well as in many subsequent stages of the analysis we use extensively the representation of the operators in the Bargmann space of analytic functions. Then it is proved that $$\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda }}$$ has compact resolvent and thus its spectrum consists of complex eigenvalues. Furthermore, $$\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda }}$$ generates a strongly continuous semigroup such that for all $$t \ge 0$$ and a constant $$c > 0$$ the estimate $$\displaystyle {\mid \mid e^{-t\mathbb {H}_{\lambda ', \mu , \lambda }}\mid \mid \le e^{ct}}$$ holds, and the operator $$\displaystyle { e^{-t(\mathbb {H}_{\lambda ', \mu , \lambda } + c)}}$$ is compact for all $$t > 0$$. Moreover, the solutions of the Cauchy problem $$\displaystyle {\frac{du}{dt} + \mathbb {H}_{\lambda ', \mu , \lambda }u = 0}$$ can be expanded as a series in the eigenvectors of $$\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda }}$$. Similar results concerning the associated semigroups are established for the operator $$\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda }}$$ for $$\lambda ' = 0$$. If $$p = 0$$, $$\lambda ' = 0$$ and $$\lambda \in i\mathbb {R}$$ then $$\displaystyle { \mathbb {H}_{\lambda ', \mu , \lambda }}$$ is the displaced harmonic oscillator. Secondly If $$p = 1$$, the Reggeon field theory (Boreskov et al. in Phys Atomic Nucl 69(10):1765–1780, 2006; Gribov in Sov. Phys. JETP 26(2):414–423, 1968) is governed by $$\displaystyle { \mathbb {H}_{\lambda ', \mu ,\lambda }}$$. In this case for $$\lambda '> 0 ,\mu > 0$$, let $$\sigma (\lambda ',\mu , \lambda ) \ne 0$$ be the smallest eigenvalue of $$H_{\lambda ',\mu ,\lambda }$$, we show in this paper that $$\sigma (\lambda ',\mu , \lambda )$$ is positive, increasing and analytic function on the whole real line with respect to $$\mu$$ and that the spectral radius of $$H_{\lambda ',\mu ,\lambda }^{-1}$$ converges to that of $$H_{0,\mu ,\lambda }^{-1}$$ as $$\lambda '$$ goes to zero. Hence we can exploit the structure of $$H_{\lambda ',\mu ,\lambda }^{-1}$$ as $$\lambda '$$ goes to zero to deduce the main results of Ando–Zerner established (Ando and Zerner in Commun Math Phys 93:123–139, 1984) on the function $$\sigma (0,\mu , \lambda )$$. Thirdly, If $$\lambda ' = \mu = 0$$, we consider the generalized operator $$\displaystyle {H^{p,m} = a^{*^{p}} (a^{m} + a^{*^{m}})a^{p}}$$; $$(p, m=1, 2,\ldots )$$ of $$\displaystyle {-\frac{i}{\lambda }\mathbb {H}_{0, 0, \lambda }}$$ acting on Bargmann space. For this operator, we find some conditions on the parameters p and m for that $$\displaystyle { H^{p,m}}$$ to be completely indeterminate. It follows from these conditions that $$\displaystyle { H^{p,m}}$$ is entire of the type minimal. And by applying the main result of the authors (Intissar and Intissar in Complex Anal Oper Theory 11(3):491–505, 2017), we show that $$\displaystyle { H^{p,m}}$$ and $$\displaystyle { H^{p,m}+ H^{*^{p,m}}}$$ are connected at the chaotic operators (where $$H^{*^{p,m}}$$ is the adjoint of the $$H^{p,m}$$).

## Keywords

Spectral theory Gribov–Intissar’s operators Semigroups Unbounded chaotic operators Entire operators Bargmann space Reggeon field theory

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